Chemical group theory : techniques and applications

256 Pages·1995·11.4 MB·English

by  Danail Bonchev, D.H. Rouvray

CHEMICAL
GROUP THEORY
TECHNIQUES AND
APPLICATIONS
D. BONCHEV and
D.H. ROUVRAY
GORDON AND BREACH PUBLISHERS
CHEMICAL GROUP THEORY
Mathematical Chemistry
A series of books edited by:
Danail Bonchev, Department of Physical Chemistry, Higher Institute of
Chemical Technology, Burgas, Bulgaria
Dermis H. Rouvray, Department of Chemistry, University of Georgia, Athens,
Georgia, USA
Volume 1
CHEMICAL GRAPH THEORY: Introduction and Fundamentals
Volume 2
CHEMICAL GRAPH THEORY: Reactivity and Kinetics
Volume 3
CHEMICAL GROUP THEORY: Introduction and Fundamentals
Volume 4
CHEMICAL GROUP THEORY: Techniques and Applications
This book is part of a series. The publisher will accept continuation orders
which may be cancelled at any time and which provide for the automatic
billing and shipping of each title in the series upon publication. Please write for
details.
CHEMICAL GROUP THEORY
Techniques and Applications
Edited by
Danail Bonchev
Department of Physical Chemistry, Higher Institute of Technology,
Burgas, Bulgaria
and
Dennis H. Rouvray
Department of Chemistry, University of Georgia, Athens, USA
GORDON AND BREACH PUBLISHERS
Australia Austria Belgium China France Germany
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Switzerland Thailand United Kingdom United States
British Library Cataloguing in Publication Data
Chemical Group Theory: Techniques and
Applications. - (Mathematical Chemistry
Series, ISSN 1049-801; Vol. 4)
I. Bonchev, Danail II. Rouvray, D. H.
III. Series
541.2015122
ISBN 2-88449-034-5
CONTENTS
Introduction to the Series vii
Preface xii
1. ALGEBRAIC TECHNIQUES FOR GROUP THEORY
D.J. Klein 1
1. Introduction 1
2. Group and Convolution Algebras 3
3. Representation Algebra 5
4. Wigner Algebra 8
5. Racah Tensor Algebra 10
6. Class and Character Algebra 13
7. Carrier Spaces 17
8. Symmetry Adaptation 19
9. Sequence Adaptation 21
10. Recoupling Coefficients 23
11. Vector Coupling 26
12. Wigner Symbols, etc. 28
13. Wigner-Eckart Theorem 30
14. Conclusion 32
15. References 33
2. COMBINATORICS AND SPECTROSCOPY
K. Balasubramanian 37
1. Introduction 37
2. Combinatorial Techniques 39
3. Applications to Magnetic Spectroscopy 43
4. Applications to Hyperfine Structure in ESR 51
5. Applications of Combinatorics to Fullerenes 58
6. References 67
3. SYMMETRY-DERIVED METHODS FOR OBTAINING
THE SPECTRA OF CHEMICALLY SIGNIFICANT GRAPHS
R.B. King 71
1. Introduction 71
2. Hiickel Theory, Graph Spectra, and the Sachs Algorithm 77
3. The General Symmetry Factoring Method:
Davidson’s Work 81
4. Specialization to Symmetry Elements of Period Two 87
5. Specialization to Symmetry Elements of Period Three 94
V
vi Contents
6. Hidden Symmetry and Subspectrality 96
7. The Ultimate in Symmetry: Distance Transitive Graphs 99
8. Examples of Symmetry Factoring 103
9. References 114
4. THE USE OF GROUP THEORY IN THE STUDY OF
NON-RIGID MOLECULES
J. Brocas 117
1. Introduction 118
2. Modes and Longuet-Higgins Groups 123
3. Van der Waals Molecules 132
4. Symmetry of Paths of Steepest Descent and
Transition States 143
5. Isomer Counting 147
6. Conclusions 157
7. References 159
5. DYNAMIC SHAPE GROUP THEORY OF MOLECULAR
NUCLEAR POTENTIALS
P.G. Mezey 163
1. Introduction 163
2. Review of the Shape Group Method of
Molecular Range NUPCO Shape Analysis 167
3. Shape Groups of External Envelope and
Internal Core Surface of Molecular NUPCO
Surfaces for Families of Nuclear Configurations 178
4. Summary 187
5. References 188
6. GROUP THEORY AND THE GLOBAL FUNCTIONAL SHAPES
FOR MOLECULAR POTENTIAL ENERGY SURFACES
M.A. Collins and K.C. Thompson 191
1. Introduction 191
2. Invariant Theory 199
3. Invariant Theory in Practice 202
4. Surfaces in Terms of Polynomial Invariants 213
5. Rotation-Inversion Invariants 220
6. Invariance by Interpolation over an Orbit 226
7. Concluding Remarks 231
8. References 231
Index 235
INTRODUCTION TO THE SERIES
The mathematization of chemistry has a long and colorful history extending
back well over two centuries. At any period in the development of chemistry
the extent of the mathematization process roughly parallels the progress of
chemistry as a whole. Thus, in 1786 the German philosopher Immanuel Kant
observed [1] that the chemistry of his day could not qualify as one of the
natural sciences because of its insufficient degree of mathematization. It was
not until almost a century later that the process really began to take hold. In
1874 one of the great pioneers of chemical structure theory, Alexander Crum
Brown (1838-1922), prophesied [2] that chemistry will become a branch of
applied mathematics; but it will not cease to be an experimental science.
Mathematics may enable us retrospectively to justify results obtained by experi­
ment, may point out useful lines of research and even sometimes predict
entirely novel discoveries. We do not know when the change will take place, or
whether it will be gradual or sudden....” This prophecy was soon to be
fulfilled. Indeed, even before these words were uttered, combinatorial methods
were being employed for the enumeration of isomeric species [3]. During Crum
Brown’s lifetime algebraic equations were used to predict the properties of sub­
stances, calculus was employed in the description of thermodynamic and
kinetic behavior of chemical systems, and graph theory was adapted for the
structural characterization of molecular species.
In the present century the applications of mathematics have come thick
and fast. The advent of quantum chemistry in the 1920s brought in its wake a
host of mathematical disciplines that chemists felt obliged to master. These
included several areas of linear algebra, such as matrix theory and group theory,
as well as calculus. Group theory has become so widely accepted by chemists
that it is now used routinely in areas such as crystallography and molecular
structure analysis. Graph theory seems to be following in the footsteps of group
theory and is currently being exploited in a wide range of applications involving
the classification, systemization, enumeration and design of systems of chemical
interest. Topology has found important applications in areas as diverse as the
characterization of potential energy surfaces, the discussion of chirality, and the
description of catenated and knotted molecular species. Information theory has
yielded valuable insights into the nature of thermodynamic processes and the
origin of life. The contemporary fascination with dissipative systems, fractal
phenomena and chaotic behavior has again introduced new mathematics, such
as catastrophe theory and fractal geometry, to the chemist.
viii Introduction to the Series
All of these and numerous other applications of mathematics that have
been made in the chemical domain have brought us to a point where we con­
sider it may now be fairly said that mathematics plays an indispensable role in
modern chemistry. Because of the burgeoning use of mathematics by chemists
and the current feeling that mathematics is opening up some very exciting new
directions to explore, we believe that the 1990s represent a particularly auspi­
cious time to present a comprehensive treatment of the manifold applications
of mathematics to chemistry. We were persuaded to undertake this somewhat
awesome task after much reflection and eventually decided to publish our
material in a series of volumes, each of which is to be devoted to a discussion
of the applications of a specific branch of mathematics. The title of our series,
Mathematical Chemistry, was chosen to reflect as accurately as possible the
proposed contents. The term ‘mathematical chemistry’ was coined in the early
1980s to designate the field that concerns itself with the novel and nontrivial
application of mathematics to chemistry. Following the usual practice in this
area, we shall interpret chemistry very broadly to include not only the tradi­
tional disciplines of inorganic, organic and physical chemistry but also their
hybrid offspring such as chemical physics and biochemistry.
It is anticipated that each of the volumes in our series will contain five to
six separate chapters, each of which will be authored by a leading expert in the
respective field. Whenever it is evident that one such volume is insufficient to
do justice to a wealth of subject matter, additional volumes devoted to applica­
tions of the same branch of mathematics will be published. In this way it is
hoped that our coverage will indeed be comprehensive and reflect significant
developments made up to the end of the twentieth century. Our aim will be not
only to provide a background survey of the various areas we cover but also to
discuss important current issues and problems, and perhaps point to some of
the major trends that might be reasonably expected in mathematical chemistry
in the early part of the new millennium. In the first few volumes of our series
we propose to examine the applications to chemistry of graph theory, group
theory, topology, combinatorics, information theory and artificial intelligence.
It may be of interest to observe here that mathematical chemists have
often applied and even sought after branches of mathematics that have tended
to be overlooked by the chemical community at large. This is not to imply that
the mathematics itself is necessarily new - in fact, it may be quite old. What is
new is the application to chemistry; this is why the word novel was employed
in our earlier definition of mathematical chemistry. The thrill of discovering
and developing some novel application in this sense has been an important
source of motivation for many mathematical chemists. The other adjective used
in our definition of mathematical chemistry, i.e. nontrivial, is also worthy of
brief comment. To yield profitable new insights, the mathematics exploited in a
chemical context usually needs to be of at least a reasonably high level. In an
endeavor to maintain a uniformly high level, we shall seek to ensure that all of
the contributions to our volumes are written by researchers at the forefront of
Introduction to the Series ix
their respective disciplines. As a consequence, the contents of our various
volumes are likely to appeal to a fairly sophisticated audience: bright under­
graduate and postgraduate students, researchers operating at the tertiary level
in academia, industry or government service, and perhaps even to newcomers
to the area desirous of experiencing an invigorating excursion through the
realms of mathematical chemistry. Overall, we hope that our series will provide
a valuable resource for scientists and mathematicians seeking an authoritative
and detailed account of mathematical techniques to chemistry.
In conclusion, we would like to take this opportunity of thanking all our
authors, both those who have contributed chapters so far and those who have
agreed to submit contributions for forthcoming volumes. It is our sincere hope
that the material to be presented in our series will find resonance with our
readership and afford many hours of enjoyable and stimulating reading.
Danail Bonchev
Dennis H. Rouvray
1. I. Kant, Metaphysische Anfangsgrunde der Naturw is sens draft, Hartknoch
Verlag, Riga, 1786.
2. A. Crum Brown, Rept. Brit. Assoc. Adv. Sci., 45-50, 1874.
3. F.M. Flavitsky, J. Russ. Chem. Soc., 3, 160, 1871.